Is proton + antiproton $\rightarrow$ $\pi^0$ meson allowed?

In the center-of-mass frame of the collision, the total momentum is zero. Therefore, the neutral pion must be produced at rest in this frame, or else the interaction violates momentum conservation. But the neutral pion's mass is lower than the mass of the proton and antiproton, so producing it at rest violates energy conservation. Therefore, the interaction is forbidden, as is any interaction of the form

$$A+B\to C$$

where $m_C<m_B+m_A$.

It's hard to know what you're asking. Protons, of course, are not fundamental particles; a proton is $(u u d)$ and a $\pi^0$ is $\frac{1}{\sqrt{2}}(u \bar{u} - d\bar{d})$, so you can't have $p\, \bar{p} \to \pi^0$ with no other products.

However, $\pi^0$s are produced all the time in $p\, \bar{p}$ collisions such as those in the LHC, so it is certainly possible.